The amplitude in the non-relativistic low-energy limit can be used to estimate the bending angle of a light scalar field under the influence of a heavy scalar. In order to do that let us consider only the non-analytic part of the 1-loop amplitude in the graviton double-cut, where we subtract the complex contribution since it is not contributing to the scattering angle as pointed out in [1]}, [2]}, which in the small momentum transfer limit \(s\approx -\mathbf {q}^2\)  has the form:
\(i\mathcal {M}^{(g)}(\mathbf {q})&=&i\mathcal {M}^{(0)}+i\mathcal {C}^{(g)}_{0\ n.a.} +\frac{i \kappa ^4E^3m^3}{16\pi }\frac{1}{\mathbf {q}^2}\log \biggl (\frac{\mathbf {q}^2}{m^2}\biggr ) \ , \nonumber \\& = &E^2 m^2\biggl [-\frac{\kappa ^2}{\mathbf {q}^2}-\kappa ^4\frac{15}{512\pi ^2}\frac{m}{\vert \mathbf {q}\vert }\nonumber \\& &-\kappa ^4\frac{3}{128\pi ^2}\log ^2\biggl (\frac{\mathbf {q}^2}{\bar{\mu }^2}\biggr )+\kappa ^4\frac{b_u}{(8\pi )^2}\log \biggl (\frac{\mathbf {q}^2}{\bar{\mu }^2}\biggr )\nonumber \\ \ & &+\kappa ^4\log \biggl (\frac{\mathbf {q}^2}{m^2}\biggr )\biggl (\frac{1}{512\pi ^2}-\frac{3}{64\pi ^2}\log \biggl (\frac{\bar{\mu }^2}{4E^2}\biggr )\biggr )\biggr ] \ ,\) 
